Gaokao Exam Gaokao Mathematics: Functions and Calculus 7

Question 1

If f(x) = x² and the tangent at point P has slope 4, find the coordinates of P.

Accepted Answer

(2, 4)

Answer

f'(x) = 2x = 4 → x = 2; f(2) = 4; P = (2, 4).

Question 2

Evaluate ∫₋₁¹ (x³ + x) dx.

Accepted Answer

0

Answer

f(x) = x³ + x is an odd function; integral of any odd function over a symmetric interval [-a, a] is 0.

Question 3

The function f(x) = e^(2x) - 2e^x + 1. The minimum value of f on ℝ is:

Accepted Answer

0

Answer

Let u = eˣ > 0; f = u² - 2u + 1 = (u-1)²; minimum = 0 when u = 1, i.e., eˣ = 1 → x = 0.

Question 4

If f''(a) = 0 and f'''(a) ≠ 0, then x = a is:

Accepted Answer

An inflection point

Answer

f''(a) = 0 with f'''(a) ≠ 0 means the concavity changes at x = a, making it an inflection point.

Question 5

Let f(x) = x^(1/3). Find f'(8).

Accepted Answer

1/12

Answer

f'(x) = (1/3)x^(-2/3); f'(8) = (1/3)·8^(-2/3) = (1/3)·(1/4) = 1/12.

Gaokao Exam Practice Test